Find the value of x.

The absolute value equation that has the solution set with digits -18, 0, and X on a number line is :

|X| = X if X ≥ 0

|X| = -X if X < 0

An equation is a mathematical statement that shows the equality of two expressions. It typically includes one or more variables (represented by letters) and may also include constants, coefficients, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation.

If the solution set for the absolute value equation has digits -18, 0, and X on a number line, then the equation must have two separate cases: one where X is positive and one where X is negative. We can write the absolute value equation as:

|X| = X for X ≥ 0

|X| = -X for X < 0

To include the values -18 and 0 in the solution set, we need to check each case separately:

|X| = X if X is greater than 0 or zero. The equation X = a must therefore be resolved, where an is either -18 or 0. As X 0 is satisfied by only one value in the solution set, the case's solution is X = 0.

Example 2: X < 0

When X is negative, |X| equals -X. So, we must find a, where an is either -18 or 0, to solve the equation -X = a. This case's solution is X = 18, as it is the only value in the solution set that satisfies the condition X 0.

As a result, the absolute value equation whose answer is represented by the numbers -18, 0 and X on a number line is:

|X| = X if X ≥ 0

|X| = -X if X < 0

or

|X| = 0 if X = 0

|X| = 18 if X < 0

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Using equations,

8. The original dimensions of the rectangle are:

Length = 6m and Width = 2m.

9. The meter contains the following:

Value of nickels = 0.63

Value of dime = 2.26

Value of quarters = 0.21

10. Volume of the sphere is V. So, the value of r will be:

r = ∛ (3v/4π)

11. Solving for the linear equation we can get the value of m = -1

A mathematical statement that has two expressions with equal values separated by the symbol "equal to" is called an equation.

In the question,

1. Let the length of the rectangle be = x.

Let the width of the rectangle be = w.

Given,

l = 3w

When the width is increased by 4 meters, (w + 4) meters is the width now.

Then, the rectangle becomes a square.

That means length and width are equal.

l = w+4

Using l from the 1st equation,

3w = w + 4

Subtracting w from both sides,

⇒ 3w - w = 4

⇒ 2w = 4

Diving both sides by 2,

⇒ w = 2 meters.

We know, l = 3w

l = 3 × 2

l = 6 meters.

2. Here,

Let nickels be n.

Let dimes be d.

Let quarters be q.

Given,

Triple the number of quarters as nickels, so:

3q = n

Solving this for q,

q = n/3

Dimes is one than double the times of nickels, so:

d = 2n + 1

Total value of coins = $3.10

So, n + d + q = 3.10

Substituting the value of q and d,

⇒ n + 2n + 1 + n/3 = 3.10

⇒ 3n + n/3 = 3.10 - 1

⇒ (9n+n)/3 = 2.10

⇒ 10n = 6.3

⇒ n =0.63.

We can now find,

d = 2n + 1

= 2 × 0.63 + 1

= 1.26 + 1

= 2.26

q = n/3

= 0.63/3

= 0.21

3. Given,

Volume of sphere,

V = 4/3 × π × r³

Solving for r,

Multiply 3/4 on both sides,

⇒ 3v/4 = π r³

Divide both sides by π,

⇒ 3v/4π = r³

Now, taking cube root on both sides we get,

r = ∛ (3v/4π)

4. Now given a linear equation,

[tex]y=-2x^{m+2}[/tex]

As the equation is a linear equation,

The value of (x, y) we can take as (0,0)

So, 1 = m +2

⇒ m = -1

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To find the equation of a line parallel to y = x - 5 that goes through (7,0), we need to use the fact that parallel lines have the same slope.

The slope of the line y = x - 5 is 1, since the coefficient of x is 1. Therefore, the slope of the parallel line we want to find is also 1.

Using the point-slope form of a line, we can write the equation of the parallel line as:

y - y1 = m(x - x1)

where (x1, y1) is the point (7,0) and m is the slope of the line, which we know is 1.

Plugging in the values, we get:

y - 0 = 1(x - 7)

Simplifying, we get:

y = x - 7

Therefore, the equation of the line parallel to y = x - 5 that goes through (7,0) is y = x - 7.

Answer:

y = x - 7

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = x - 5 ← is in slope- intercept form

with slope m = 1

• Parallel lines have equal slopes , then

y = x + c ← is the partial equation

to find c substitute (7, 0 ) into the partial equation

0 = 7 + c ( subtract 7 from both sides )

- 7 = c

y = x - 7 ← equation of parallel line

The findings of the research project suggest that regular exercise can be an effective tool for managing stress in university students.

A sample research project is given below:

Study of Data: The impact of exercise on stress levels in university students

Prediction: We predict that there will be a significant decrease in stress levels following a four-week exercise program.

Data Collection: Participants were recruited through flyers and social media. They were asked to complete a pre-test stress survey before beginning the exercise program, and a post-test stress survey after completing the program. The exercise program consisted of a combination of cardiovascular and resistance training, meeting three times a week for four weeks.

Table of Figures:

Participant Pre-Test Stress Score Post-Test Stress Score

         1                          32                                 20

         2                         28                                  17

         3                         36                                  25

         ...                         ...                                   ...

         30                          41                                  22

Measures of Central Tendency:

Graph: Line graph showing the change in stress levels over time, with pre-test scores in blue and post-test scores in red.

Written Analysis: The line graph shows a consistent decrease in stress levels across all participants following the four-week exercise program. The majority of participants showed a decrease in stress levels of 10 or more points, indicating a significant reduction in stress. There is a clear pattern of improvement in stress levels over time, with the post-test scores consistently lower than the pre-test scores.

Findings: Our study found a significant decrease in stress levels following a four-week exercise program in university students. Our prediction was accurate, as we anticipated a decrease in stress levels.

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we use the definition of supplementary angles again to conclude that ∠3 and ∠5 are supplementary.

To prove that ∠3 and ∠5 are supplementary, we can use the following steps:

Given: w ∥ x and y is a transversal.

∠1 ≅ ∠5 by the corresponding angles postulate.

By definition, ∠3 and ∠1 are a linear pair, so they are supplementary.

Therefore, m∠3 + m∠1 = 180 by the definition of supplementary angles.

Substituting m∠5 for m∠1, we get m∠3 + m∠5 = 180.

Therefore, by the definition of supplementary angles, ∠3 and ∠5 are supplementary.

In step 1, we use the corresponding angles postulate, which states that when two parallel lines are cut by a transversal, the pairs of corresponding angles are congruent. In step 2, we use the definition of a linear pair, which states that when two angles form a line, they are supplementary. In step 3, we use the definition of supplementary angles, which states that the sum of two supplementary angles is 180 degrees. In step 4, we substitute m∠5 for m∠1, since we know that ∠1 ≅ ∠5. Finally, in step 5, we use the definition of supplementary angles again to conclude that ∠3 and ∠5 are supplementary.

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Answer:

The vertex of the quadratic function f(x) = -2x^2 + 8x is (2, 8).

Step-by-step explanation:

To find the vertex of the quadratic function f(x) = -2x^2 + 8x, we first need to find the x-coordinate of the vertex using the formula:

x = -b / 2a

where a = -2 and b = 8. Substituting these values, we get:

x = -8 / 2(-2) = 2

Now, to find the y-coordinate of the vertex, we can substitute this value of x back into the equation: f(2) = -2(2)^2 + 8(2) = 8

The area of the shape made of two rectangles is 85 square cm.

To find the area of the shape made of two rectangles with dimensions 9cm, 5cm, 8cm, and 5cm, follow these.

As per the given information,
Rectangle 1 has dimensions 9cm and 5cm.
Rectangle 2 has dimensions 8cm and 5cm.
For calculating the area of each rectangle using the formula:

Area = length × width
[tex]{Area\: of\: Rectangle} _1[/tex] = 9cm × 5cm = 45 square cm
[tex]{Area of Rectangle}_ 2[/tex] = 8cm × 5cm = 40 square cm
Add the areas of both rectangles to find the total area of the shape.
Total Area = [tex]{Area\: of\: Rectangle} _1[/tex] + [tex]{Area\: of\: Rectangle} _2[/tex]
Total Area = 45 square cm + 40 square cm

Total Area = 85 square cm.

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Question: The shape below is made of two rectangles joined together. 5 cm 9 cm 8 cm 5 cm Find the total area of the shape.

Answer:

Base length=22

width=30

height=15

volume of pyramid= Lxwxh/3

=22x 30x 15/3 =3300in^3.

The missing length is 8m which is option A.

To find the missing length, we need to use the formula for the area of a rectangle:

area = length x width

We are given the area as 114 m² and the width as 15 m. Let's substitute these values into the formula:

114 = length x 15

Solving for length, we get:

length = 114/15

length = 7.6

Therefore, the missing length is approximately 7.6 m.

However, none of the given answer choices match this value exactly. The closest answer is A. 8 m, which is off by only 0.4 m. Therefore, A. 8 m is the best answer choice.

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Based on the lengths of the sides, we can see that the triangle is scalene. Based on the slopes of the sides, we can see that the triangle is acute since all the slopes are negative or less than 1.

A triangle is a three-sided polygon with three vertices. The triangle's internal angle, which is 180 degrees, is constructed.

To find the desired slopes and lengths of the triangle formed by the points M(-7, 8), N(-1, 9), and O(0, 3), we can use the distance formula and the slope formula.

Distance formula:

The distance between two points (x₁, y₁) and (x₂, y₂) is given by:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Using this formula, we can find the lengths of the sides of the triangle:

Length of MO:

d(MO) = √((0 - (-7))² + (3 - 8)²) = √(7² + 5²) = √(74)

Length of NO:

d(NO) = √((-1 - (-7))² + (9 - 8)²) = √(6² + 1²) = √(37)

Length of MN:

d(MN) = √((-1 - (-7))² + (9 - 8)²) = √(6² + 1²) = √(37)

Slope formula:

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by:

m = (y₂ - y₁) / (x₂ - x₁)

Using this formula, we can find the slopes of the sides of the triangle:

Slope of MO:

m(MO) = (3 - 8) / (0 - (-7)) = -5/7

Slope of NO:

m(NO) = (9 - 8) / (-1 - (-7)) = 1/6

Slope of MN:

m(MN) = (9 - 8) / (-1 - (-7)) = 1/6

Characterization of the triangle:

Based on the lengths of the sides, we can see that the triangle is scalene (no two sides have the same length). Based on the slopes of the sides, we can see that the triangle is acute (all angles are less than 90 degrees) since all the slopes are negative or less than 1. Additionally, we can see that the side MO is the longest side of the triangle, and since the slope of MO is negative, we can conclude that the angle opposite side MO is the largest angle in the triangle.

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Answer:

x=[tex]\frac{5b-a}{3-b}[/tex]

Step-by-step explanation:

3x+a=b(x+5)

3x+a=bx+5b

3x-bx=5b-a

x(3-b)=5b-a

x=[tex]\frac{5b-a}{3-b}[/tex]

Answer:

Step-by-step explanation:

The p-value is  0.046, or 4.6% if a sample of shoppers showed stating that the supermarket brand was as good as the national brand.

To calculate the p-value, we use null and alternative hypotheses, as well as the test statistic and its distribution under the null hypothesis.

By the null hypothesis, we can say that supermarket brands are as good as national brands, and by the alternative hypothesis, we can say that supermarket brands are not as good as national brands.

This hypothesis can be tested by performing a two-tailed z-test for proportions.

In a sample of n shoppers, say that x shoppers say that supermarket brands are as good as domestic brands. The sample percentage can be calculated as follows:

p hat = x/n

in the null hypothesis, the  sample proportion is equal to the hypothesized proportion, that is 0.5

The test statistic for a two-sided z-test for proportions is given by:

z = (p-hat - p0) / sqrt(p0(1-p0)/n)

where p0 is the hypothesized proportion under the null hypothesis.

In this case, we have:

p-hat = 0.5

p0 = 0.5

n = sample size

The null hypothesis states that the supermarket brand is as good as the national brand, so we would expect the proportion of shoppers who state this to be 0.5.

If the test statistic z falls in the rejection region (i.e., if |z| > 1.96 for a significance level of 0.05),

we would reject the null hypothesis and conclude that there is evidence to suggest that the supermarket brand is not as good as the national brand.

If the test statistic falls in the non-rejection region (i.e., if |z| <= 1.96 for a significance level of 0.05),

we would fail to reject the null hypothesis and conclude that there is insufficient evidence to suggest that the supermarket brand is not as good as the national brand.

To calculate the p-value, we need to find the probability of getting a test statistic as extreme or more extreme than the one we observed, assuming the null hypothesis is true.

For a two-tailed test, the p-value is twice the observed area of ​​the tails above the absolute value of the test statistic.

Assuming the sample size is large enough, the distribution of the test statistic can be approximated as a standard normal distribution under the null hypothesis.

Suppose in a sample of 100 shoppers, 50 said that supermarket brands are as good as domestic brands. after that:

p hat = 0.5

p0 = 0.5

n=100

The test statistic is:

z = (p hat - p0) / sqrt(p0(1-p0)/n) = 0 / sqrt(0.5 * 0.5 / 100) = 0

The p-value is the probability that the test statistic is extreme or more extreme than 0. This is the probability of getting further away from the 50/100 ratio or 0.5 in either direction.

p-value = P(p hat <= 0.4 or p hat >= 0.6) = 2 * P(p hat <= 0.4) = 2 * P(Z <= ( 0.4 - 0.5) / sqrt (0.5 * 0.5 / 100) ) = 2 * P(Z <= -2) ≈ 0.046

where Z is a standard normal random variable.

Therefore, the p-value is approximately 0.046, or 4.6%.

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Answer and Explanation:

The value of a brand new car,

P

=

$

28000

The value depreciates every year,

r

=

19

%

Time,

t

=

1

4

Number of compounding period (Compounded quarterly),

n

=

4

Write a function to represent the value of the car after

t

years:

A

=

P

(

1

−

r

n

)

n

×

t

∴

A

=

28000

(

1

−

19

%

4

)

4

t

Let us evaluate the value of the car after one quarter:

A

=

28000

(

1

−

19

%

4

)

4

t

=

28000

(

1

−

19

100

4

)

4

×

1

4

=

28000

(

1

−

0.19

4

)

4

×

1

4

=

28000

(

1

−

0.19

4

)

1

=

28000

(

1

−

0.19

4

)

=

28000

(

1

−

0.0475

)

=

28000

(

0.9525

)

∴

A

=

26670

Therefore, the value of the car after one quarter is

$

26670

.

To calculate the percentage rate of change per quarter:

Percentage rate

=

P

−

A

P

×

100

%

=

28000

−

26670

28000

×

100

%

=

1330

28000

×

100

%

=

70

⋅

19

70

⋅

400

×

100

%

=

19

400

×

100

%

=

0.0475

×

100

%

∴

Percentage rate of change per quarter

=

4.75

%

Hence, the percentage rate of change per quarter is

4.75

%

.

Two parents are present in 69% of all families with children ages six to eleven. Option d is correct.

Option d is the correct choice, which states that 69 percent of all families with children aged six to 11 have two parents. This information is supported by various studies and statistics, which have consistently shown that the majority of families in the United States consist of two-parent households. While single-parent families do exist, they are less common than families with two parents.

This information is important in understanding the dynamics and structure of families in the US, as well as in developing policies and programs that support and strengthen families.

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--The complete question is, According to your text, 69 percent of all families with children aged six to 11:

a) are harmonious and stable.

b) have one parent.

c) are chronically poor.

d) have two parents.
Link to the text is,  open.maricopa.edu/devpsych/chapter/chapter-9-early-adulthood/--

When you put together 8 copies of the unit fraction 1/2, you are essentially adding 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2.

To add fractions, you need to make sure the denominators are the same. In this case, all the fractions already have the same denominator of 2.

So, we can add the numerators together:

1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 = 8/2

Now, we can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 2:

8/2 = 4

Therefore, putting together 8 copies of the unit fraction 1/2 gives us a fraction of 4/1, or simply 4.

The problem involves a line segment divided into two parts in a specific ratio. The ratio of the length of the lesser part to the length of the greater part is found to be (√2 - 1).

Let the length of the whole line segment be x, and let y be the length of the greater part. Then the length of the lesser part is (x - y).

According to the problem statement, the ratio of the lesser part to the greater part is the same as the ratio of the greater part to the whole. Mathematically, we can write this as:

(x - y)/y = y/x

Simplifying this equation, we get:

x^2 - y^2 = y^2

x^2 = 2y^2

Taking the square root of both sides, we get:

x = y√2

Therefore, the value of the ratio of the lesser part to the greater part is:

(x - y)/y = (√2 - 1)

So, the answer will be (√2 - 1).

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Answer:

you need an equally

Step-by-step explanation:

The statement which is "The binomial distribution table, the number of trials must be greater than 30." is the not correct statement related to the binomial and Poisson distributions. Hence option a is the right choice.

a. "When using the binomial distribution table, the number of trials must be greater than 30"This is an incorrect statement.

There is no such rule stating that the number of trials must be greater than 30 for the binomial distribution.

b. "The Poisson distribution represents the random arrival of events per unit of time or space" This is the correct statement.

The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, given a constant average rate of occurrence.

c. "Only two outcomes are possible in a binomial situation" This is correct statement.

A binomial distribution represents the number of successes in a fixed number of trials, each with only two possible outcomes (success or failure).

d. To use the Poisson distribution tables, mu must be known or be able to be calculated: This is correct.

In order to use the Poisson distribution tables, the mean number of events (represented by the symbol µ) must be known or calculable.

So, option a is right choice.

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Question :-

Which of the following is not a correct statement about the binomial and Poisson distributions?

a. When using the binomial distribution table, the number of trials must be greater than 30.

b. The Poisson distribution represents the random arrival of events per unit of time or space.

c. Only two outcomes are possible in a binomial situation.

d. To use the Poisson distribution tables, Mu must be known or be able to be calculated

Simon needs approximately 48.6 gallons of gas for his trip from Orlando, FL to Sacramento, CA.

To calculate the number of gallons of gas Simon needs for his trip, you can use the following formula:

Gallons = Miles / Miles per gallon

First, convert the distance of 2,804 miles to miles per gallon by dividing by the car's average mileage per 2

gallons:2,804 / 48.6 = 57.75 mpg

Then, use this value to calculate the number of gallons of gas Simon needs for his trip:

Gallons = Miles / Miles per gallon

Gallons = 2,804 / 57.75

Gallons ≈ 48.6 gallons

Therefore, Simon needs approximately 48.6 gallons of gas for his trip from Orlando, FL to Sacramento, CA.

The answer should be rounded to the nearest tenth, so the final answer is 48.6 gallons.

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If -4 is a zero with multiplicity 1, then (x + 4) is a factor of the polynomial. Similarly, if 1 is a zero with multiplicity 2, then (x - 1)^2 is a factor of the polynomial. Therefore, we can write the polynomial in factored form as:

f(x) = a(x + 4)(x - 1)^2

where "a" is a constant that we need to determine.

To find "a", we use the fact that f(0) = -12. Substituting x = 0 into the equation above, we get:

f(0) = a(0 + 4)(0 - 1)^2

-12 = -4a

Solving for "a", we get:

a = 3

Therefore, the polynomial is:

f(x) = 3(x + 4)(x - 1)^2

Note that this polynomial has a zero at x = -4 (with multiplicity 1), a zero at x = 1 (with multiplicity 2), and f(0) = -12.

Step-by-step explanation:

This is the equation for a AREA greater than or equal to a line

2x-10 >= 5y

y <= 2/5 x -2      there are infinite answers

Bianca practiced 28 hours a day whereas Filipe practiced 12 fewer hours than Bianca, if Filipe practiced for 16 hours

below are the situations and the operations used

Rising =  +

difference = -

to find the part of the grand total= ÷

Sharing  or Grouping = *

Since "Filipe trained 12 hours less than Bianca" and "Filipe trained 16 hours", we can use rising(+) to find the amount of time Bianca trained.

it is calculated by using the following formula

Bianca's training time = Filipe's training time + 12

Bianca practice hours = 16 + 12

Bianca Practice Hours = 28

So Bianca practiced 28 hours a day.

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the correct answer is: Wide Awake, with a median value of 4.5 gallons. Thus, option A is correct.

Based on the data given, the correct measure of center to determine which shop typically sells the most amount of coffee per hour would be the median.

Calculating the medians for both coffee shops:

For Coffee Ground:

[tex]1.5, 12, 11, 9.5[/tex] (sorted data)

Median [tex]= (11 + 12)/2 = 11.5[/tex]

For Wide Awake:

[tex]2.5, 18, 3, 6[/tex] (sorted data)

Median [tex]= (3 + 6)/2 = 4.5[/tex]

Comparing the medians, we can see that Wide Awake has a median value of 4.5 gallons, which is higher than the median value of 11.5 gallons for Coffee Ground.

Therefore, the correct answer is: Wide Awake, with a median value of [tex]4.5[/tex] gallons.

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Answer: Coffee Ground typically sells more coffee per hour than Wide Awake.

Step-by-step explanation:

Given that:

Coffee ground :

Hour 1: 1.5

Hour 2: 20

Hour 3: 3.5

Hour 4: 12

Hour 5: 2

Hour 6: 5

Hour 7: 11

Hour 8: 7

Hour 9: 2.5
We add up all of these numbers and divide by the total number of hours to find the mean:

(1.5 + 20 + 3.5 + 12 + 2 + 5 + 11 + 7 + 2.5) / 9 = 7.222

therefore mean is 7.2

Median:

Arranging them in Ascending or descending order, we get:

1.5,2,2.5,3.5,5,7,11,12,20

As we know, The median of discrete data is :

((n+1)/2)th element if the number of elements is odd

Average of (n/2) th and ((n+1)/2)th element if the number of elements is even. the value

Here the number of elements is odd, so the median is:

((n+1)/2)th =(9+1)/2 = 5th element

i.e., the median is 5.

Wide Awake:

Given that:

Hour 1: 2.5

Hour 2: 10

Hour 3: 4

Hour 4: 18

Hour 5: 4

Hour 6: 3

Hour 7: 6.5

Hour 8: 5

Hour 9: 15

We add up all of these numbers and divide by the total number of hours to find the mean:

(2.5 + 10 + 4 + 18 + 4 + 3 + 6.5 + 5 + 15) / 9 = 7.056

therefore mean is 7.1

Median:

Arranging them in Ascending or descending order, we get:

3,6.5,5,4,4,2.5,10,18,15

Here the number of elements is odd, so the median is:

((n+1)/2)th =(9+1)/2 = 5th element

i.e., the median is 4.

As we can observe, the mean and median of coffee ground is high.

So we can conclude that the Coffee Ground shop has sold more coffee.

Answer:

Yes, the length of the intercepted arc will have the same measurement. When a central angle intercepts an arc of a circle, the measure of the angle is equal to the measure of the intercepted arc. In this case, both angles ABC and DEF are right angles, which means that they each intercept half of the circle. Therefore, the length of the intercepted arcs will be the same.

As a result, the slope-intercept form equation for the line through points (5, 21) and (11, 3) is:

y = -3x + 36

Therefore, the equation of the line through (4, 8) and (12, 20) in slope-intercept form is:

y = (3/2)x + 2

The slope of the line must first be determined using the following formula to determine the equation of a line through two points in slope-intercept form:

slope[tex](m) =\frac {(y2 - y1)}{(x2 - x1)}[/tex]

where the coordinates for two points are (x1, y1) and (x2, y2).

The line's slope-intercept form is thus applicable, and it is as follows:

y = mx + b

where b is the y-intercept and m is the slope that we just discovered.

We possess

slope [tex](m) = \frac{(3-21)}{ (11-5)}[/tex],

m = -3

So, using one of our points, we can determine b:

y-21=-3(x-21)

As a result, the slope-intercept form equation for the line through points (5, 21) and (11, 3) is:

y = -3x + 36

9) (4, 8) and (12, 20)

The slope is

[tex]m= \frac{20-8}{12-4}\\m=\frac{12}{8}=\frac{3}{2}[/tex]

equation is (y-8)=1.5(x-4)

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Answer: The missing side e is approximately 21.1 units long. So, the correct answer is E. None of the above

Answer:

d

Step-by-step explanation:

12 - 9.5 = 2.5

Answer:

Step-by-step explanation:

[tex]-\frac{8}{35}[/tex]

Answer:

B and C are the answers according to the expression